Visualization of Local Deformation in Metallic Materials through In Situ X-Ray Diffraction Mapping

Abstract

We propose a novel method for the visualization of the shape and stress–strain evolution in the local deformation of metallic materials. We combined in situ X-ray diffraction with simultaneous transmittance measurements while scanning the specimen, to visualize the temporal evolution of the spatial distributions of thickness, stress, and inhomogeneous strain in the sample. Through a comparison of stress and strain distributions within the local deformation regions, it was demonstrated that work hardening is the dominant factor contributing to the increase in stress in the local deformation area of pure copper. This method is expected to contribute to the elucidation of local deformation in manufacturing processes.

1. Introduction

Metallic materials such as copper and aluminum are widely used in the manufacturing of electrical wires and terminals because of their superior balance between strength and conductivity. In automobiles, the increasing amount of wiring owing to the advancement of electrical appliances necessitates conductor downsizing to maintain or reduce the total weight of the wires. This requires the development and use of electric wires with simultaneously higher strength and higher conductivity than currently used wires, necessitating elucidation of the atomic-level structural changes resulting from manufacturing processes and deformation during service.

Recently, in situ X-ray diffraction (XRD) has been employed for the analysis of atomic-level deformation behavior during tensile tests. Particularly, synchrotron radiation has been used to make highly accurate and time-resolved measurements. These experiments revealed the propagation of dislocations in metallic materials such as aluminum, copper, nickel, nickel–titanium alloy, and steel [1–10].

However, local deformation occurring in the later stage of deformation is difficult to assess using conventional in situ XRD. In general, during tensile tests, local deformation occurs after uniform deformation in certain areas where the specimen becomes thinner. In these local deformation regions, stress becomes concentrated, leading to further deformation. Therefore, a spatial distribution of deformation arises within the tensile specimen. This local deformation is difficult to evaluate using conventional in situ XRD owing to the lack of information about the spatial distribution, because measurement area is limited within the specimen. Although there have been attempts to combine digital image correlation with in situ XRD to calculate strain distributions from images [11–13], digital image correlation has its own limitations, such as the inability to accurately measure deformation in the thickness direction.

Therefore, there are still many unknowns regarding the correlation among deformation, stress distribution, and strain distribution in local deformation. In contrast, as deformation occurs nonuniformly in many manufacturing processes, elucidation of the spatial and temporal evolution of thickness, stress, and strain during local deformation is important for product development.

Therefore, we propose a novel measurement technique for the visualization of the shape and stress–strain evolution in metallic materials. Herein, we report the results obtained using this method on a copper plate.

2. Experimental Procedure

2.1 Materials

A rolled copper plate with a thickness of 0.3 mm was used. Its detailed chemical composition is given in Table 1. To investigate the grain size and strain of the specimen, electron beam backscatter diffraction (EBSD) measurements were performed using Oxford Symmetry mounted on ZEISS Gemini 450. The step size in EBSD was 0.05 µm.

Table 1 Chemical composition of the test specimen (mass%).

2.2 Measurement

The experimental settings employed in the measurements is schematically shown in Fig. 1(a).

Fig. 1

(a) Schematic of in situ XRD mapping during the tensile test; (b) magnified view of the tensile specimen. (online color)

Measurements were performed in SPring-8 BL16XU. The incident X-ray was monochromated to 37 keV using a Si (111) double-crystal monochromator, and higher harmonics were removed using a Rh-coated mirror. The beam was shaped into a 0.1 mm square using a quadrant slit.

Diffraction from the sample were measured in a transmission mode using a two-dimensional detector (Pilatus 300K CdTe, Dectris Ltd.). In this experiment, only one of the three modules of the detector was used. The detector was positioned at 445 mm from the sample, with the long side of the detection area oriented horizontally in the tensile direction. Employing this setting, we obtained five reflections characteristic for fcc metals: 111, 200, 220, 311, and 222. Additionally, the intensity of transmitted X-rays was simultaneously measured using a photodiode.

Tensile tests were conducted using the Linkam 10073B stage at room temperature. This tensile stage was mounted on an automated stage with two-axis motion capabilities (vertical and horizontal), enabling repetitive mapping measurements. Figure 1(b) shows a magnified view of tensile specimen fabricated in a dumbbell shape via laser machining, with the tensile axis parallel to the TD. The length and width of parallel section was 1 mm. The green line in Fig. 1(b) represents the window area of the tensile-testing machine. X-rays can pass through only in this area. In the repetitive mapping experiment, the measured area was larger than the window size, with dimensions of 5.6 mm × 2.0 mm. The time required for each mapping measurement was 513 s. The specimen was pulled at the same velocity from both sides at the elongation rate of 2.56 × 10⁻⁵ s⁻¹.

3. Results and Discussions

3.1 EBSD

Figure 2 shows inverse pole figure map of the as-received tensile test specimen in the ND–TD plane. No prominent crystal orientation is observed, and the average grain size is approximately 9.9 µm.

Fig. 2

Inverse pole figure map of the as-prepared tensile test specimen in the ND–TD plane. (online color)

3.2 Tensile test

Figure 3 shows the stress–strain curve. The solid line shows nominal stress and nominal strain, indicating a uniform elongation of around 16%. The true stress was calculated from the nominal stress and nominal strain under the assumption of constant volume. This assumption is typically applicable only in the uniform deformation region. However, herein, it was extended to the entire region for comparison with the stress calculated from XRD, which will be discussed later. In Fig. 3, the dashed line shows true stress and true strain, indicating the tensile strength of 250 MPa.

Fig. 3

Results of the tensile test. The solid line is plotted using nominal stress and nominal strain, and the dashed line using true stress and true strain.

3.3 Transmission map

Figure 4(a1) exhibits the X-ray transmission map at the pretensile stage (0 s). Figure 4(a2) shows the map before fracture (7183–7695 s), corresponding to a nominal strain range of 18.4%–19.7%. Figure 4(a3) presents the map after fracture (7696–8208 s), corresponding to a nominal strain range of 19.7%–21.0%. In the figure, the left–right direction is the tensile direction, and the units of transmittance are given in percent (%). The areas with no X-ray transmission along the outer area represent the shadow of the tensile stage, while the regions with approximately 20% transmittance in the center correspond to the tensile specimen. The areas with transmittance close to 100% are air. In Fig. 4(a1), the specimen exhibits uniform transmittance. In Fig. 4(a2), a slight increase in transmittance is observed in the central region. Furthermore, in Fig. 4(a3), the transmittance in the central region is 100%, indicating the occurrence of fracture. In the transmittance map, pixels with low transmittance values were considered to show the tensile-testing machine, while pixels with high transmittance values were assumed to represent air. In the following analysis, only pixels with transmittance between 18.5% and 60% were considered.

Fig. 4

X-ray transmission maps at the (a1) pretensile, (a2) before-fracture, and (a3) after-fracture stages; thickness maps at the (b1) pretensile, (b2) before-fracture, and (b3) after-fracture stages. (online color)

3.4 Analysis procedure

In this experiment, we calculated the thickness, stress, and strain maps. First, thickness maps were calculated from the transmittance map and X-ray absorption coefficient, which is determined by the composition, its density, and X-ray energy. X-ray transmission map, Tr(x, z), was computed as follows:

  

\begin{equation} \mathit{Tr}(x,z) = \exp[-\mu t(x,z)] \end{equation} (1)

where t(x, z) is the thickness map, and μ is the X-ray linear absorption coefficient. Therefore, t(x, z) can be calculated using the following equation:

  

\begin{equation} t(x,z) = -\frac{\ln[\mathit{Tr}(x,z)]}{\mu} \end{equation} (2)

Herein, we adopted a value of μ = 5.49 mm⁻¹, obtained from the NIST database [14].

Second, stress maps were calculated from lattice parameter change obtained via XRD. In this experiment, single-component materials were measured at room temperature; therefore, all changes in lattice constants were attributed to stress. In the analysis, curve fitting was performed using the nonlinear least-squares method with pseudo-Voigt function [15]. Lattice constants a_hkl were calculated for each of the five diffraction angles using the following equations:

  

\begin{equation} 2d_{hkl}\sin \theta = \lambda \end{equation} (3)

  

\begin{equation} \frac{1}{d_{hkl}{}^{2}} = \frac{h^{2} + k^{2} + l^{2}}{a_{hkl}^{2}} \end{equation} (4)

Equation (3) is the Bragg law, where d_hkl is the spacing of (hkl) planes, θ is the half of diffraction angle (2θ), and λ is the wavelength. Equation (4) is the crystallographic relation in a cubic system. To obtain the lattice constants used for stress calculations, these five lattice constants were plotted against the Nelson–Riley function as the x-axis and extrapolated to 2θ = 180° [16].

At the earlier stage of tensile test, deformation is uniform. Therefore, the lattice parameters of the specimen are considered uniform, and their average values can be directly correlated with the global stress obtained from the load cell. As shown in Fig. 5, a linear relationship is observed between the lattice parameter of the specimen and the true stress calculated from the load cell at five time steps after the start of the test. As an additional note, the initial point having a negative value corresponds to a slight compression that occurred when installing the specimen onto the tensile-testing machine. The obtained regression equation is σ = 60831a − 220245. The stress map was calculated by applying this regression equation to the entire region. The advantage of this stress calculation method is that it does not require the knowledge of material’s properties such as Young’s modulus and can be calculated based solely on the information obtained within this measurement. Notably, if the temperature or composition change, the stress can be determined by obtaining a regression equation between the deviation from the circular shape of Debye ring.

Fig. 5

True stress calculated from the load cell at five time steps after the start of the test against the lattice parameter. The dashed line is the regression line. (online color)

Third, inhomogeneous strain ε was obtained via classical Williamson–Hall method [17, 18]:

  

\begin{equation} \frac{\beta \cos \theta}{\lambda} = 4\varepsilon \frac{\sin \theta}{\lambda} + \frac{1}{D} \end{equation} (5)

where β is the integral width, and D is the crystallite size. The changes in orientation and phase transformation rates can be visualized using the intensity ratio between multiple peaks.

3.5 Results

Figure 4(b1) shows the thickness map for the initial stage, indicating that the thickness of the specimen is uniform. Figure 4(b2), (b3) shows that the central area of the specimen becomes thinner, indicating the occurrence of local deformation. These maps can be used to represent the changes in the three-dimensional shape of the specimen.

Figure 6 shows the stress and inhomogeneous strain maps. Figure 6(a1) and 6(b1) exhibit the stress and inhomogeneous strain maps in the pretensile stage. Stress and inhomogeneous strain are low overall but highly dispersed, which was attributed to the poor particle statics.

Fig. 6

Stress maps at the (a1) pretensile, (a2) before-fracture, and (a3) after-fracture stages; inhomogeneous strain maps at the (b1) pretensile, (b2) before-fracture, and (b3) after-fracture stages. (online color)

Figures 6(a2) and 6(b2) show the stress and inhomogeneous strain maps immediately before the fracture. The stress and inhomogeneous strain increase in areas corresponding to the thin regions in Fig. 4(b2). The variations in values between adjacent pixels are lower than in Figs. 6(a1) and 6(b1). This was attributed to a reduction in grain size and improvement in particle statics with the advancement of deformation. The arrow in Fig. 6(a2) indicates the point of maximum stress (596 MPa), which is more than twice higher than the tensile strength shown in Fig. 3. The spatial distributions of heterogeneous strain and stress show similarities.

Figure 6(a3) exhibits the stress map immediately after fracture. The stress is completely relieved, while the inhomogeneous strain remains in the material (Fig. 6(b3)). These results demonstrate that the proposed method can simultaneously and separately analyze both elastic and plastic behavior.

Figure 7 shows the Kernel Average Misorientation (KAM) map in the ND–TD plane obtained through EBSD. The KAM values were calculated as the average values of points within 10° from adjacent pixels. Figure 7(a) shows the KAM map of the as-prepared tensile test specimen, with the strain being extremely low. Figures 7(b) and 7(c) exhibit KAM maps of the specimen after in situ XRD mapping. The fractured specimen was cut along the dashed line in Fig. 6(b3) and observed from the RD direction at positions indicated by the arrows in Fig. 6(b3). Strain increases as the position gets closer to the fracture position, consistent with Fig. 6(b3).

Fig. 7

KAM map in the ND–TD plane. (a) KAM map of the as-prepared tensile test specimen. (b) and (c) KAM maps at positions indicated by the arrows in Fig. 6(b3). (online color)

Figure 8 shows the overlay of the average stress from stress map and the stress at the maximum position in Fig. 6(a2) on the global stress measured by the load cell. The values obtained from the map before the tensile test were plotted at t = 0 s, while the values obtained from the map during tensile test were plotted at the midpoint of measuring time of each map. Average value of each stress map corresponds to the true stress obtained using the load cell, confirming the validity of the employed stress calculation method. The local stress at the point of maximum position is significantly larger than the global stress. This result demonstrates that the proposed method can evaluate strength characteristics in local deformation more accurately than the conventional approach of using averages of load-cell measurements.

Fig. 8

Overlay of the average stress from stress map and the stress at the maximum position from Fig. 6(a2) on the stress–strain curve measured by the load cell.

Figure 9 compares various parameters between pixels with large deformation and small deformation. The pixel with large deformation (x = −0.45 mm, z = 0.2 mm) is represented by hollow symbols, while the pixel with small deformation (x = 1.55 mm, z = 0 mm) is represented by filled symbols. Figure 9(a) shows the change in local thickness over time. In the pixel with small deformation, the thickness remains constant, while in the pixel with large deformation, the thickness decreases with time. Figure 9(b) shows variations in local inhomogeneous strain with respect to local thickness. In the pixel with large deformation, heterogeneous strain increases with decreasing local thickness. At the same time, in the pixel with small deformation, the thickness remains constant and only particle-statistical variations in heterogeneous strain are observed and there is no significant increase as the pixel with large deformation. Figure 9(c) shows the variation in local stress with respect to local thickness, which is similar to the trend in heterogeneous strain in Fig. 9(b). Figure 9(d) exhibits the change in local stress with respect to local strain. In the pixels with small deformation, only variations owing to poor particle statics are observed. However, in the pixels with large deformation, stress increases almost linearly with inhomogeneous strain. This indicates that in the local deformation point of the present sample, work hardening is the dominant factor contributing to the increase in stress.

Fig. 9

Comparisons of parameters between pixels with large deformation and small deformation. The pixel with large deformation (x = −0.45 mm, z = 0.2 mm) is represented by hollow symbols, while the pixel with small deformation (x = 1.55 mm, z = 0 mm) is represented by filled symbols. (a) Change in local thickness over time. (b) Change in local inhomogeneous strain with respect to local thickness. (c) Change in local stress with respect to local thickness. (d) Change in local stress with respect to local strain.

4. Conclusion

Herein, we proposed a technique for the visualization of the shape and stress–strain evolution in metallic materials. In the local deformation point of the present sample, work hardening was the dominant factor contributing to the increase in stress. However, even in the same material (copper was used herein), the results may vary depending on factors such as grain size, coexisting elements and their amounts, and the mode of deformation. This remains a research question for future investigation.

Acknowledgments

The authors would like to thank Japan Synchrotron Radiation Research Institute (JASRI) for the approval to perform the XRD experiments at SPring-8 (Proposal No.: 2021A5031, 2021B5031, 2023A5031 and 2023B5031).

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